A woman on a bridge 84.7 m high sees a raft floating at a constant speed on the river below. She drops a stone from rest in an attempt to hit the raft. The stone is released when the raft has 8.02 m more to travel before passing under the bridge. The stone hits the water 4.86 m in front of the raft. Find the speed of the raft.
To find the speed of the raft, we can break down the problem into two parts: the vertical motion of the stone and the horizontal motion of the raft.
1. Vertical Motion:
The stone falls freely from rest under the influence of gravity. We can use the equations of motion to determine the time taken for the stone to fall and hit the water.
We know that the height of the bridge is 84.7 m and the stone hits the water 4.86 m in front of the raft, so the total distance traveled by the stone vertically is 84.7 + 4.86 = 89.56 m.
We can use the equation of motion for vertical motion:
h = ut + (1/2)gt^2
Where:
h = vertical distance (89.56 m)
u = initial vertical velocity (0 m/s)
t = time taken
g = acceleration due to gravity (-9.8 m/s^2, since it acts downwards)
Substituting the values into the equation, we get:
89.56 = 0*t + (1/2)(-9.8)t^2
Rearranging the equation and solving for t using the quadratic formula:
4.9t^2 = 89.56
t^2 = 18.26
t ≈ √18.26
t ≈ 4.27 seconds (taking the positive root)
So, the stone takes approximately 4.27 seconds to hit the water.
2. Horizontal Motion:
Now, let's focus on the horizontal motion of the raft. We know that the raft covers a distance of 8.02 m while the stone is falling. We need to find the time it takes for the raft to cover this distance.
Using the equation of motion for horizontal motion:
s = ut
Where:
s = horizontal distance (8.02 m)
u = initial horizontal velocity (unknown)
t = time taken (unknown)
Substituting the values into the equation, we have:
8.02 = u*t
Now, we have two equations with unknowns t and u:
Equation 1: t = 4.27 (from vertical motion)
Equation 2: 8.02 = u*t
Solving Equation 2 for u:
u = 8.02/t
u = 8.02/4.27
u ≈ 1.88 m/s
Therefore, the speed of the raft is approximately 1.88 m/s.